Optical scanning apparatus and scanning image forming lens

ABSTRACT

An optical scanning apparatus includes a light source for outputting light, a first lens system arranged to receive the light output from the light source and to transmit a light flux therefrom, an optical deflector arranged to receive the light flux from the first lens system and having a deflecting reflective plane to deflect the light flux from a surface therefrom and a second lens system arranged to receive the light flux deflected from the optical deflector and to condense the deflected luminous flux into an optical beam spot on a surface to be scanned so as to form images having image heights, the luminous flux condensed by the second lens system into the optical beam spot including an optical beam waist. The second lens system has a scanning and image forming element including at least one surface including a plurality of portions each having a non-arc shape in a sub-scanning direction such that at least two of the non-arc shapes are different from each other and such that an effective writing width W and a width Fs of the sub-scanned image-surface curvature located within the effective writing width satisfies the condition Fs/W&lt;0.005. Alternatively, the second lens system includes a scanning and image forming element including at least one surface including a plurality of portions each having a non-arc shape in a sub-scanning direction such that at least one lens surface is a sub non-circular arc surface and a shape in a sub-scanning cross section of the sub non-arc circular shape is a non-arc shape and the non-arc shape changes in accordance with a position of the sub-scanning cross section in the main scanning direction.

The present application is a division of co-pending U.S. applicationSer. No. 09/369,612, filed Aug. 6, 1999, pending, the teaching of whichis incorporated herein in its entirety by reference. In addition, Thepresent application is related to and hereby incorporates by referencethe subject matter of commonly assigned U.S. patent application Ser. No.09/344,633 filed on Jun. 25, 1999, entitled “OPTICAL SCANNING APPARATUS”by Seizo SUZUKI et al.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to optical scanning apparatuses andscanning image forming lenses.

2. Discussion of the Background

The required writing density of an optical scanning device used in arecording unit of a facsimile or digital copying machine, a laserprinter or other similar image forming apparatuses, has been recentlygreatly increased. Also, there has been a great demand recently forensuring that the light beam spots converged on a scanned surface aregreatly reduced in size and have a uniform size.

In order to obtain an excellent recorded image having a constant imageresolution by use of an image forming apparatus having an opticalscanning apparatus, it is important that a beam spot diameter does notvary in relation to a scanning position, i.e., that the beam spots areuniform in diameter along a scanning line. To realize such stability ofbeam spot diameter, an image-surface curvature in an optical systemneeds to be satisfactorily corrected, and various attempts to realizethe satisfactory correction of the image-surface curvature have beenmade.

To perform a satisfactory image writing at a remarkably high writingdensity such as 600 dpi or 1200 dpi, a beam spot having a small beamspot diameter is necessary. To realize the small beam spot diametersatisfactorily and with good stability, a conventional geometric opticalcorrection of image-surface curvature, optical magnification or the likeis insufficient, and it is important to make sure that a wave-opticalwave aberration is constant irrespective of the image height of the beamspot. More specifically, as the beam spot diameter decreases, a largerluminous-flux diameter becomes necessary, so that paraxial correction bygeometrical optics is insufficient.

In the prior art, a polygon mirror functioning as a light deflectingdevice is sometimes tilted to achieve a desired surface tilt foreffectively deflecting a luminous flux to be impinged on a surface to bescanned. However, this surface tilt of the polygon mirror causesproblems with a field of curvature and beam waist of the beam spotimpinged on the surface to be scanned. Prior art devices have notrecognized the combined problems with field of curvature and beam waistand therefore, have not provided solutions for correcting theseproblems. As most, the prior art only focused on correcting the beamwaist problem only at the center image height and did not recognize thenecessity or the way to correct the beam waist problem for all imageheights.

Also, when multiple beams are used for a light source, the image surfacecurvature causes uneven scanning pitches in the sub-scanning direction.That is, the scanning line intervals are not constant over an entirerange of image heights, and instead, the scanning line intervals varyaccording to image height.

SUMMARY OF THE INVENTION

In order to overcome the problems described above, preferred embodimentsof the present invention provide an optical scanning apparatus and ascanning image forming lens which correct wave aberrationsatisfactorily, achieve a reliably uniform and greatly reduced beam spotdiameter, while achieving a desired very high writing density.

In addition, preferred embodiments of the present invention provide anoptical scanning apparatus and a scanning image forming lens whichcorrect and minimize a beam waist at all image heights along the surfaceto be scanned and also correct field of curvature such that a beam spotdiameter is minimized and made uniform for all beam spots impinged onthe surface to be scanned.

According to one preferred embodiment of the present invention, anoptical scanning apparatus includes a light source for outputting light,a first lens system arranged to receive the light output from the lightsource and to transmit a light flux therefrom, an optical deflectorarranged to receive the light flux from the first lens system and havinga deflecting reflective plane to deflect the light flux from a surfacetherefrom and a second lens system arranged to receive the light fluxdeflected from the optical deflector and to condense the deflectedluminous flux into an optical beam spot on a surface to be scanned so asto form images having image heights, the luminous flux condensed by thesecond lens system into the optical beam spot including an optical beamwaist. The second lens system has a scanning and image forming elementincluding at least one surface including a plurality of portions eachhaving a non-arc shape in a sub-scanning direction such that at leasttwo of the non-arc shapes are different from each other and such that aneffective writing width W and a width Fs of the sub-scannedimage-surface curvature located within the effective writing widthsatisfies the condition Fs/W<0.005.

In another preferred embodiment of the present invention, an opticalscanning apparatus includes a light source for outputting light, a firstlens system arranged to receive the light output from the light sourceand to transmit a light flux therefrom, an optical deflector arranged toreceive the light flux from the first lens system and having adeflecting reflective plane to deflect the light flux from a surfacetherefrom and a second lens system arranged to receive the light fluxdeflected from the optical deflector and to condense the deflectedluminous flux into an optical beam spot on a surface to be scanned so asto form images having image heights, the luminous flux condensed by thesecond lens system into the optical beam spot including an optical beamwaist. The second lens system includes a scanning and image formingelement including at least one surface including a plurality of portionseach having a non-arc shape in a sub-scanning direction such that atleast one lens surface is a sub non-circular arc surface and a shape ina sub-scanning cross section of the sub non-arc circular shape is anon-arc shape and the non-arc shape changes in accordance with aposition of the sub-scanning cross section in the main scanningdirection.

It should be noted that the preferred embodiments of the presentinvention are intended for use with apparatuses having a writing densityof about 1200 dpi, 2400 dpi or greater than 2400 dpi. However, thepreferred embodiments of the present invention may also be used withapparatuses having a writing density of about 300 dpi to 600 dpi.

The second lens system may preferably include either one lens element ortwo lens elements or more than two lens elements.

At least one surface of the lens element or lens elements of the secondlens system is preferably non arc shape in a main scanning direction.

The light source in the apparatus of preferred embodiments of thepresent invention may be a single light beam source or a multiple beamsource.

Other features, elements, advantages and improvements of the presentinvention will become evident in the following detailed description ofpreferred embodiments of the present invention with reference to theattached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the present invention and many of theattendants advantages thereof will be readily obtained as the samebecomes better understood by reference to the following detaileddescription thereof when considered in conjunction with accompanydrawings, wherein:

FIG. 1 is a schematic drawing illustrating a construction of an opticalscanning image-forming apparatus according to a preferred embodiment ofthe present invention;

FIG. 2 is a schematic drawing illustrating a construction of anotherscanning image-forming apparatus according to another preferredembodiment of the present invention;

FIG. 3 is a diagram showing a non-arc shape in the main scanningdirection used in at least one lens of the optical scanningimage-forming apparatus according to preferred embodiments of thepresent invention;

FIG. 4 is a diagrams illustrating image-surface curvature and a constantvelocity characteristic of a preferred embodiment of the presentinvention;

FIGS. 5(a) and 5(b) are diagrams illustrating depth curves of a beamspot diameter (variation the beam spot diameter relative to thedefocusing of the beam spot) of a preferred embodiment of FIG. 4 in themain and sub-scanning directions, respectively;

FIG. 6 is a diagram illustrating image-surface curvature and constantvelocity characteristic of another preferred embodiment of the presentinvention;

FIGS. 7(a) and 7(b) diagrams illustrating depth curves of a beam spotdiameter (variation of beam spot diameter relative to the defocusing ofthe beam spot) of another preferred embodiment of FIG. 6 in the main andsub-scanning directions, respectively;

FIGS. 8(a)-8(e) are diagrams illustrating wave aberrations at variousimage heights when an aperture for shaping beams has a substantiallyrectangular opening in the preferred embodiment of FIG. 6;

FIGS. 9(a )-9(e) are diagrams illustrating wave aberrations at variousimage heights when the aperture has substantially rectangular openingwith rounded four corners in the preferred embodiment shown in FIG. 6;

FIG. 10 is a diagram illustrating changes of the paraxial curvature inthe sub-scanning cross-section of the sub non-circular arc shape in themain scanning direction in the preferred embodiment of FIG. 6;

FIG. 11 is a diagram illustrating image-surface curvature and constantvelocity characteristic of a Gird preferred embodiment; and

FIGS. 12(a) and 12(b) diagrams illustrating depth curves of a beam spotdiameter (variation of beam spot diameter relative to the defocusing ofthe beam spot) of the preferred embodiment of FIG. 6 in the main andsub-scanning directions, respectively, when the aperture has asubstantially rectangular opening with rounded four corners.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Referring now to the accompanying drawings, wherein like referencenumerals designate identical or corresponding parts throughout theseveral views, preferred embodiments of the present invention aredescribed.

A scanning image-forming optical system according to preferredembodiments of the present invention is a scanning image-forming opticalsystem in which a deflected luminous flux deflected by a lightdeflecting device having a deflection reflective surface is converged asa beam spot on a scanned surface, and has characteristics as describedbelow.

More specifically, the scanning image-forming optical system accordingto preferred embodiments of the present invention preferably includesone or more lenses, and at least one lens surface of the lenses is a subnon-circular arc surface.

According to at least one preferred embodiment of the present invention,the scanning image-forming optical system can be constituted of a singlelens in the simplest form. The scanning image-forming system may alsoinclude two or more lenses, or may include a combination of one or morelenses and one or more elements of a reflective image-forming system,for example, such as a concave mirror or suitable elements.

The meaning of main scanning direction and sub-scanning direction inthis specification refer to directions originally defined on the scannedsurface, but in the present specification, directions corresponding tothe main and sub-scanning directions along an optical path leading tothe scanned surface from a light source are also referred to as the mainand sub-scanning directions, respectively. Therefore, in a certain case,the main and sub-scanning directions are not necessarily parallel withthe main and sub-scanning directions on the scanned surface.

The sub non-circular arc surface is defined as a shape in a sub-scanningcross section that is a non-circular arc shape, and the non-circular arcshape changes in accordance with a position of the sub-scanning crosssection in the main scanning direction.

The sub-scanning cross-section indicates a virtual flat section which issubstantially perpendicular to the main scanning direction in thevicinity of a lens surface. Moreover, in the vicinity of the lenssurface, the virtual flat section coincident with a plane swept by amain beam of an ideal deflected luminous flux is referred to as the mainscanning cross-section.

The shape of the sub non-circular arc surface in the scanningimage-forming optical system is configured to correct the waveaberration in each scanned position on the scanned surface (convergedposition of the beam spot). Thereby, the optimum wave aberration can beset for each scanned position on the scanned surface.

The shape of the sub non-circular arc surface can be a non-circularshape even in the main scanning cross section. In this manner, in theentire pupil plane (luminous flux section on the sub non-circular arcsurface of the deflected luminous flux incident upon an arbitraryposition of the sub non-circular arc surface), the wave aberration issatisfactorily corrected. For the wave aberration correction in thescanning image-forming optical system according to preferred embodimentsof the present invention, when a wavelength is λ, the wave aberration ona pupil in RMS (root mean square) is preferably about 0.1 λ or less,more preferably about 0.07 λ (Marshal's diffraction limit) or less. Morespecifically, a wave aberration on a pupil in a root mean square issubstantially equal to$\sqrt{{1/n}\quad {\sum\limits_{i = 1}^{n}\left( {x_{i} - x} \right)^{2}}},$

wherein x is a height of wave aberration when an area of a beam spotwith X and Z coordinates is divided into blocks i=1 through i=n and aused wavelength is λ. In each scanned position on the scanned surface,when the wave aberration is about 0.1 λ or less in RMS on the pupil, thesatisfactory-shape, small-diameter beam spot is reliably and uniformlyachieved.

There are many different ways to achieve this desired RMS relationship.For example, this RMS feature can be achieved by: (1) setting a noncircular arc amount of a lens surface having a non arc shape to bedifferent for all image heights so as to correct wave aberration for allimage heights; (2) provide coupling lens which is constructed to correctwave aberration for image heights; (3) configure the first lens systemto correct wave aberration for all image heights. Other suitable methodsfor achieving the desired RMS relationship for correcting waveaberration for all image heights may also be used.

For achieving a very small beam spot diameter, when the beam spotdiameter of the beam spot on the scanned surface is defined with anintensity 1/e² in a line spread function of light intensity distributionin the beam spot, an image-forming function of the scanningimage-forming optical system can be set in such a manner that the spotdiameter is about 50 μm or less in an effective writing range in boththe main and sub-scanning directions.

For the line spread function, when a center coordinate of the beam spotformed on the scanned surface is used as a reference, and coordinates inthe main and sub-scanning directions arc Y, Z and the light intensitydistribution of the beam spot is f (Y, Z), the line spread function LSZin direction Z is defined with LSZ(Z)=∫f (Y, Z) dY (integration isperformed on the entire width of the beam spot in direction Y), and theline spread function LSY in direction Y is defined with LSY(Y)=∫f (Y, Z)dZ (integration is performed on the entire width of the beam spot indirection Z).

These line spread functions LSZ(Z), LSY(Y) generally have substantialGaussian distribution type shapes, and the beam spot diameters in thedirections Y and Z are given at the widths in the directions Y, Z of anarea in which these line spread functions LSZ(Z), LSY(Y) have maximumvalues of about 1/e² or more. More specifically, the beam spot diameterin direction Y, Z defined in this manner is preferably about 50 μm orless in the effective writing range, irrespective of the image height ofthe beam spot. The beam spot diameter defined by the line spreadfunctions as described above can easily be measured by scanning the beamspot via a slit at a uniform velocity, receiving light passed throughthe slit with an optical detector, and integrating a received lightamount. An apparatus for performing such measurement is well known.

It is not easy to form the desired beam spot having a beam spot diameterof about 50 μm or less only with the geometric optical image-surfacecurvature correction, but according to preferred embodiments of thepresent invention, the desired beam spot having a beam diameter of about50 μm or less is reliably achieved using one or more sub noncircular arcsurfaces in the scanning image-forming optical system.

More specifically, there are many ways to correct field of curvatureincluding: (1) provide non arc shape in the main scanning direction ineach of four surfaces of two lens elements constituting the second lenssystem; (2) provide non arc shape in the main scanning direction in eachof three surfaces of two lens elements constituting the second lenssystem; (3) correcting sub-scanning field of curvature by using aspecial toric surface for at least two lens surfaces in the second lenssystem; (4) increasing the number of lens elements. Other suitablemethods for correcting field of curvature may also be used.

In the scanning image-forming optical system according to preferredembodiments of the present invention, a lateral magnification β₀ on anoptical axis in the sub-scanning direction preferably satisfies acondition:

0.2 <|β₀|<1.5.  (1)

When an upper limit 1.5 of the condition (1) is exceeded, the lateralmagnification of the scanning image-forming optical system is increased.Therefore, when the beam spot diameter is to be reduced, an emissionpupil diameter in the sub-scanning direction becomes excessively largeand it becomes difficult to correct the wave aberration on the entirepupil. Furthermore, NA of a coupling lens for incorporating the luminousflux from the light source has to be enhanced. Moreover, environmentalvariations or image-surface position variations caused by a mountingerror of the scanning image-forming optical system tend to increase, andit becomes difficult to reduce the beam spot diameter. When a lowerlimit 0.2 is exceeded, the magnification is excessively low, an apertureopening diameter is reduced, light transfer efficiency is lowered, and ahigh-speed writing becomes difficult.

In the scanning image-forming optical system according to the presentinvention, as regards the sub-scanning direction, the lateralmagnification β₀ on the optical axis, and lateral magnification β_(h) atan arbitrary image height h preferably satisfy the following condition:

0.93 <|β_(h)/β₀|<1.07.  (2)

When the beam spot is converged, a beam waist diameter of luminous fluxvaries substantially in proportion to variations of the lateralmagnification of the scanning image-forming optical system. Therefore,in order to obtain a beam spot having a uniform and stable beam spotdiameter, it is important to make sure the lateral magnification isconstant for each image height by making sure that the condition (2) issatisfied.

In the scanning image-forming optical system according to preferredembodiments of the present invention, a non-circular arc amount of thenon-circular arc shape in the sub-scanning cross section of the subnon-circular arc surface can be changed asymmetrically relative to themain scanning direction, preferably for all image heights.

The non-circular arc amount is an amount of deviation from a circulararc (paraxial curvature radius).

When a rotating polygon mirror is used as the light deflecting device,its rotation center is deviated from the optical axis of the scanningimage-forming optical system for installation. Therefore, in accordancewith the deflection, a reflecting point in the deflection reflectivesurface changes, and a deflection starting point of the deflectedluminous flux varies, so that an optical sag occurs. When the sag ispresent, a path through which the luminous flux passes differs on sidesof plus and minus image heights of the optical axis of the scanningimage-forming optical system. Thus, the generated wave aberrationdiffers asymmetrically in accordance with the image height of the beamspot, but by arranging the non-circular arc amount of the subnon-circular arc surface of the scanning image-forming optical system soas to be asymmetric relative to the main scanning direction, theasymmetric wave aberration caused by the sag is corrected, and theoptimum wave aberration is set for each image height.

In the scanning image-forming optical system according to preferredembodiments of the present invention, the paraxial curvature in thesub-scanning cross section changes asymmetrically relative to the mainscanning direction, and at least one lens surface in which the change ofthe paraxial curvature has two or more extreme values can be provided.

The optical sag causes deterioration of the image-surface curvatureespecially of the sub-scanning direction (hereinafter referred to assub-scanned image-surface curvature), but in the scanning image-formingoptical system according to preferred embodiments of the presentinvention, by using at least one surface in which the paraxial curvaturein the sub-scanning cross section changes in the main scanningdirection, and the curvature change is asymmetric relative to the mainscanning direction and has two or more extreme values, especially thelateral magnification of the sub-scanning direction can be keptsubstantially constant over the effective writing area to achieve a beamspot which is stable in beam spot diameter while the sub-scannedimage-surface curvature is satisfactorily corrected.

Additionally, when the paraxial curvature in the sub-scanning crosssection relative to a lens height h is represented by the function C(h)relative to the lens height h, the extreme value indicates a point atwhich its one-step differential is dC(h)/dh=0, and before and afterwhich the sign of dC(h)/dh changes (maximum or minimum value is taken).

Generally, when the lateral magnification is to be kept constant, ahigh-order curved image-surface curvature is easily generated.Especially in the image-forming optical system with fewer number oflenses, a sagitall image-surface curvature is easily generated, which isrepresented by the expression “aH²+bH⁴”, H being an image height, and a,b being coefficients.

As described above, when a plane is used, in which the change of theparaxial curvature radius in the sub-scanning cross section in the mainscanning direction is provided with a plurality of extreme values, thehigh-order curved image-surface curvature is corrected by changing poweron the lens surface in a high-order manner, so that the sub-scannedimage-surface curvature can effectively be corrected.

In the scanning image-forming optical system according to preferredembodiments of the present invention, for at least one of the extremevalues in the change of the paraxial curvature in the sub-scanning crosssection, its position he of the main scanning direction and an effectivelens height hmax from a lens optical axis on a plus or minusimage-height side preferably satisfy a condition:

|(he)/(hmax)|>0.5.  (3)

An image height Hn of a maximum bulged position of the sagitallimage-surface curvature represented by “aH²+bH⁴” is Hn=(1/2) Hm=0.71 Hn,when Hm is an effective writing height (“Lens Design Technique (OpticalIndustry Technical Association)” authored by Fumio KONDO, pp.146-148).

In order to correct the bulge in the vicinity of about 0.71 times theeffective writing height Hm, it is effective to provide the vicinity ofthe lens surface position corresponding to the relevant position withthe extreme value of the curvature in the sub scanning cross section.Taking into consideration that the four-order or higher-orderimage-surface curvature is to be corrected, it is preferable that theaforementioned lie and hmax satisfy the condition (3). Additionally,hmax denotes an effective lens height on the plus image-height side whenhe>0, and an effective lens height on the minus image-height side whenhe<0.

Here, the plus image-height side means a side where the luminous fluxfrom the light source is incident upon the deflection reflectivesurface.

In the scanning image-forming optical system according to preferredembodiments of the present invention, a lens surface where the paraxialcurvature in the sub-scanning cross section changes asymmetricallyrelative to the main scanning direction and where the change of theparaxial curvature has two or more extreme values can be a subnon-circular arc surface. It should be noted that the lens surface maybe a surface other than the sub non-circular arc surface.

In the scanning image-forming optical system according to preferredembodiments of the present invention, an effective writing width W, anda width Fs of the sub-scanned image-surface curvature in the effectivewriting width preferably satisfy the following condition:

Fs/W<0.005.  (4)

The condition (4) is preferably satisfied to obtain the stablesmall-diameter beam spot by preventing variation of the sub-scannedimage-surface curvature.

In the above-mentioned scanning image-forming optical system accordingto preferred embodiments of the present invention, the shape of the subnon-circular arc surface in the main scanning cross section can bedetermined so as to correct a constant velocity characteristic of thebeam spot. It should be noted that the shape of another surface in themain scanning cross section can be configured to achieve, the objectdescribed above.

The scanning image-forming optical system according to preferredembodiments of the present invention can be an anamorphic optical systemhaving a function of placing the vicinity of the deflection reflectivesurface and the scanned surface position in a geometric opticalconjugate relationship for the sub-scanning direction. In this manner,when the scanning image-forming optical system is anamorphic, the effectof surface tilt in the light deflecting device is corrected.

The scanning image-forming optical system of preferred embodiments ofthe present invention can also include one or more lenses as describedabove, and, therefore, can include two lenses.

When the scanning image-forming optical system is made of two lenses,the number of sub non-circular arc surfaces, the degree of freedom inarrangement, and further, the degree of freedom in the shape of otherlens surfaces are greatly increased, so that desired opticalcharacteristics are easily realized.

In the scanning image-forming optical system according to preferredembodiments of the present invention, if two lenses constitute thescanning image-forming lens, the sub non-circular arc surface may beused in a lens surface on the side of a scanned surface of a lens on theside of the scanned surface, or in a lens surface on the side of thelight deflecting device of the lens on the side of the scanned surface.

The scanning image-forming optical system according to preferredembodiments of the present invention can have a function of converging adeflected luminous flux as a weak converged luminous flux in the mainscanning direction onto the scanned surface, and a function ofconverging the deflected luminous flux as a parallel luminous flux inthe main scanning direction onto the scanned surface.

An optical scanning device according to preferred embodiments of thepresent invention is an optical scanning device for deflecting aluminous flux from a light source by a light deflecting device having adeflection reflective surface, and converging the deflected luminousflux as a beam spot on the scanned surface by a scanning image-formingoptical system to perform light scanning, characterized in that any oneof the above-described scanning image-forming optical systems is used asthe scanning image-forming optical system.

In the optical scanning device according to preferred embodiments of thepresent invention, the deflected luminous flux deflected by the lightdeflecting device is a weak converged luminous flux in the main scanningdirection. Alternatively, the deflected luminous flux deflected by thelight deflecting device is a parallel luminous flux in the main scanningdirection.

As the light deflecting device, a rotating polygon mirror or a rotatingtwo-surface mirror or a rotating single-surface mirror can preferably beused. The scanning image-forming optical system according to preferredembodiments of the present invention may include a system having nosurface tilt correcting function. Therefore, in the optical scanningdevice according to preferred embodiments of the present invention, whenthe scanning image-forming optical system has no surface tilt correctingfunction, the rotating polygon mirror or the rotating two-surface mirrorhaving excellent surface precision may be preferably used, or therotating single-surface mirror having no surface tilt may be preferablyused. In this case, the luminous flux from the light source (asemiconductor laser is generally used) is preferably taken via thecoupling lens and caused to be incident upon the deflection reflectivesurface of the light deflecting device as the parallel luminous flux ora weakly convergent or divergent luminous flux. It should be noted thatthe coupled luminous flux is subjected to beam shaping by passingthrough an aperture opening.

In the optical scanning device according to preferred embodiments of thepresent invention, the scanning image-forming optical system may have asurface tilt correcting function, and the luminous flux from the lightsource may be formed into an image which is elongated in the mainscanning direction in the vicinity of the deflection reflective surfaceof the light deflecting device. To achieve this desired result, theluminous flux from the light source is taken via the coupling lens, andthe taken luminous flux (coupled luminous flux) may be formed into alinear image in the vicinity of the deflection reflective surface via acylindrical lens or a concave mirror having a concave cylinder surface.

When the optical scanning device according to preferred embodiments ofthe present invention takes the luminous flux from the semiconductorlaser as the light source via the coupling lens, shapes the luminousflux via the aperture, and forms the linear image which is elongated inthe main scanning direction in the vicinity of the deflection reflectivesurface via the linear image forming optical system, the opening shapeof the aperture for shaping the luminous flux can be configured to cutoff four corner portions of the main or sub-scanning direction of thecoupled luminous flux.

Now specific examples of preferred embodiments of the present inventionare described herein below.

In FIG. 1, a light source 10 preferably includes a semiconductor laserfor radiating a divergent luminous flux. The luminous flux radiated fromthe light source 10 is incorporated via a coupling lens 12 to form aweakly convergent luminous flux. The luminous flux is passed and shapedthrough an opening of an aperture 14, converged before entering acylindrical lens 16 as a linear image forming optical system, andconverged in a sub-scanning direction (direction that is perpendicularto the drawing) via the cylindrical lens 16. Additionally, an opticalpath is bent by a mirror 18, and the luminous flux is incident upon adeflection reflective surface 20A of a rotating polygon mirror 20 aslight deflecting device. The luminous flux reflected by the deflectionreflective surface 20A passes through lenses 22, 24 as the deflectedluminous flux deflected at a constant velocity, accompanied by aconstant velocity rotation of the rotating polygon mirror 20. The lenses22, 24 constitute a scanning image-forming optical system to convergethe deflected luminous flux as a beam spot on a scanned surface 26. Thebeam spot optically scans an effective writing width W of the scannedsurface 26 at a constant velocity. The scanned surface 26 issubstantially a photosensitive surface of a photoelectric photosensitivebody.

In this example of preferred embodiments of the present invention, ofthe lenses 22, 24 constituting the scanning image-forming opticalsystem, a sub non-circular arc surface is used in a lens surface of thelens 24 on the side of the scanned surface, to correct a waveaberration.

In FIG. 2 drawn in the same manner as FIG. 1, a divergent luminous fluxradiated from the light source 10 preferably comprising a semiconductorlaser is formed into a parallel luminous flux via a coupling lens 11.The luminous flux is passed and shaped through an opening of an aperture13 before entering a cylindrical lens 15, and converged in thesub-scanning direction via the cylindrical lens 15. Additionally, anoptical path is bent by the mirror 18, and the luminous flux is incidentupon a deflection reflective surface 19A of a rotating polygon mirror19. The luminous flux reflected by the deflection reflective surface 19Aforms a deflected luminous flux deflected at a constant velocity inaccordance with a constant velocity rotation of the rotating polygonmirror 19 to pass through lenses 21, 23 constituting a scanningimage-forming optical system. The lenses 21, 23 converge the deflectedluminous flux as a beam spot on the scanned surface 26, and theconverged beam spot optically scans the scanned surface 26 at a constantvelocity.

In this example of preferred embodiment, of the lenses 21, 23constituting the scanning image-forming optical system, the subnon-circular arc surface is used in a lens surface of the lens 23 on theside of the rotating polygon mirror, to correct the wave aberration.Additionally, numeral 25 denotes soundproof glass provided at a windowof a casing surrounding the rotating polygon mirror to muffle a rotatingsound of the rotating polygon mirror 19.

More specifically, an optical scanning device according to preferredembodiments shown in FIG. 1 (FIG. 2) is preferably an optical scanningdevice which is arranged to deflect a luminous flux from the lightsource 10 via the light deflecting device 20 (19) having the deflectionreflective surface 20A (19A), and to converge the deflected luminousflux as a beam spot on the scanned surface 26 via the scanningimage-forming optical system 22, 24 (21, 23) to perform light scanning,in which the scanning image-forming optical system 22, 24 (21, 23)includes one or more lenses, at least one lens surface is a subnon-circular arc surface, and a shape of the lens surface is set tosatisfactorily correct the wave aberration in each scanned position onthe scanned surface 26.

Moreover, the optical scanning device shown in FIG. 1 (FIG. 2) is anoptical scanning device for forming the luminous flux from the lightsource 10 into a linear image which is elongated in the main scanningdirection, deflecting the luminous flux via the light deflecting device20 (19) having the deflection reflective surface 20A (19A) in thevicinity of the linear image forming position, and converging thedeflected luminous flux as the beam spot on the scanned surface 26 bythe scanning image-forming optical system 22, 24 (21, 23) to performlight scanning (claim 20), in which the scanning image-forming opticalsystem 22, 24 (21, 23) is an anamorphic optical system having a functionof placing the vicinity of the deflection reflective surface 20A (19A)and the position of the scanned surface 26 in a geometric opticalconjugate relationship relative to the sub-scanning direction, andincludes two lenses.

Furthermore, in the preferred embodiment shown in FIG. 1, the deflectedluminous flux deflected by the light deflecting device 20 is a weakconverged luminous flux in the main scanning direction, the scanningimage-forming optical system 22, 24 has a function of converging thedeflected luminous flux as the weak converged luminous flux in the mainscanning direction onto the scanned surface, and the lens surface on theside of the scanned surface of the lens 24 on the side of the scannedsurface is a sub non-circular arc surface.

In the preferred embodiment shown in FIG. 2, the deflected luminous fluxdeflected by the light deflecting device 19 is a parallel luminous fluxin the main scanning direction, the scanning image-forming opticalsystem 21, 23 has a function of converging the deflected luminous fluxas the parallel luminous flux in the main scanning direction onto thescanned surface, and a lens surface on the side of the light deflectingdevice of the lens 23 on the side of the scanned surface is a subnon-circular arc surface.

Three examples of preferred embodiments will now be described.

The shape of the lens surface is specified by following equations.

Coaxial Non-spherical Surface

Lens height: represented by a depth difference between an optical axis(H=0) and H.

More specifically, the coaxial non-spherical surface is represented infollowing equation (A) using a paraxial curvature radius R, coneconstant K, and high-order coefficients A₄, A₆, . . .

X=(H²/R)/[1+{1−(1+K)(H/R)²}]+A₄·H⁴+A₆·H⁶+A₈·H⁸+ . . .   (A)

Non-Circular Arc Shape in Main scanning cross section

A depth X in an optical-axis direction is represented in the followingpolynomial equation (B) using a paraxial curvature radius Rm in the mainscanning cross section, distance Y from the optical axis in the mainscanning direction, cone constant Km, and high-order coefficients A₁,A₂, A₃, A₄, A₅, A₆ . . .

X=(Y²/Rm)/[1+{1−(1+Km)(Y/Rm)²}]+A₁·Y+A₂·Y²+A₃·Y³+A₄·Y⁴+A₅·Y⁵+A₆·Y⁶+ . ..   (B)

In the equation (B), when numeric values other than zero are substitutedin odd-order A₁, A₃, A₅ . . . , the shape becomes asymmetric in the mainscanning direction.

Curvature in sub-scanning cross section

When the curvature changes within the sub-scanning cross section in themain scanning direction (coordinate with the optical axis position beingan origin is represented by Y), the following equations (C) and (D)result. The equation (C) represents curvature Cs (Y), and (D) representscurvature radius Rs (Y). Additionally, Rs (0) represents a curvatureradius on the optical axis in the sub-scanning cross section.

Cs(Y)={1/Rs(0)}+B₁·Y+B₂·Y²+B₃·Y³+B₄·Y⁴+B₅Y⁵·+. . .   (C)

Rs(Y)=Rs(0)+B₁·Y+B₂·Y²+B₃·Y³+B₄·Y⁴+B₅·Y⁵ +. . .   (D)

In the equations (C), (D), when numeric values other than zero aresubstituted in odd-order coefficients of Y, B₁, B₃, B₅, . . . , thecurvature (or the curvature radius) in the sub-scanning cross sectionbecomes asymmetric in the main scanning direction.

Sub Non-Circular Arc Surface

The sub non-circular arc surface is represented in equation (E) using aposition Y of sub-scanning cross section in the main scanning direction,and coordinate Z of the sub-scanning direction.

X=(Y²/Rm)/[1+{1−(1+Km)(Y/Rm)²}]

+A₁·Y+A₂·Y²+A₃·Y³+A₄·Y⁴+A₅·Y⁵+ . . .

+(Z²·Cs)/[1+{1−(1+Ks)(Z·Cs)²}

+(F₀+F₁·Y+F₂·Y²+F₃·Y³+F₄·Y⁴+ . . . )·Z

+(G₀+G₁·Y+G₂·Y²+G₃·Y³+G₄·Y⁴+ . . . )·Z²

+(H₀+H₁·Y+H₂·Y²+H₃·Y³+H₄·Y⁴+ . . . )·Z³

+(I₀+I₁·Y+I₂·Y²+I₃·Y³+I₄·Y⁴+ . . . )·Z⁴

+(J₀+J₁·Y+J₂·Y²+J₃·Y³+J₄·Y⁴+ . . . )·Z⁵

+(K₀+K₁·Y+K₂·Y²+K₃·Y³+K₄·Y⁴+ . . . )·Z⁶

+(L₀+L₁·Y+L₂·Y²+L₃·Y³+L₄·Y⁴+ . . . )·Z⁷

+(M₀+M₁·Y+M₂·Y²+M₃·Y³+M₄·Y⁴+ . . . )·Z⁸

+(N₀+N₁·Y+N₂·Y²+N₃·Y³+O₄·Y⁴+ . . . )·Z⁹

+(O₀+O₁·Y+O₂·Y²+O₃·Y³+O₄·Y⁴+ . . . )·Z¹⁰

+. . .   (E)

Here, Cs represents Cs(Y) defined in the equation (C).

Moreover, Ks is defined by following equation (F).

Ks=Ks(0)+C₁·Y+C₂·Y²+C₃·Y³+C₄·Y⁴+C₅·Y⁵+ . . .   (F)

When numeric values other than zero are substituted in odd squarecoefficients of Y, B₁, B₃, B₅in equation (C) of Cs (Y), the change ofthe curvature radius in the sub-scanning cross section becomesasymmetric in the main scanning direction. In the same manner, whennumeric values other than zero are substituted in F₁, F₃, F₅ . . . , G₁,G₃, G₅ . . . and the like, the non-circular are amount in thesub-scanning cross section becomes asymmetric in the main scanningdirection.

More specifically, as described above, the sub non-circular arc surfaceis a surface in which the shape in the sub-scanning cross section is anon-circular arc, and the non-circular arc shape in the sub-scanningcross section changes in accordance with the position of thesub-scanning cross section in the main scanning direction, but in theequation (E), first and second lines on the right side indicate theshape in the main scanning cross section with the function only of thecoordinate Y of the main scanning direction. Moreover, for third andsubsequent lines on the right side, when the coordinate Y of thesub-scanning cross section is determined, the coefficient of each-orderterm is univocally determined, and the non-circular arc shape in thesub-scanning cross section in the coordinate Y is determined.

Analytic representations of the coaxial non-spherical surface,non-circular arc shape in the main scanning cross section, curvature inthe sub-scanning cross section, and sub non-circular arc surface are notlimited to the above, various representations are possible, and thesurface shape in the present invention is not limited to therepresentations defined by the above equations.

EXAMPLE 1

First, Example 1 is a specific example of the preferred embodiment shownin FIG. 1.

The luminous flux from the light source 10 or semiconductor laser iscoupled by the coupling lens 12 to form a weakly convergent luminousflux. Assuming that the weakly convergent luminous flux fails to besubjected to refraction action of an optical element on the opticalpath, a natural light-converging position by the convergence of theluminous flux is referred to as natural light-converging point. InExample 1, the position of the natural light-converging point is in aposition of about 700 mm toward the scanned surface from the deflectionreflective surface, when the deflected luminous flux is transmittedtoward the image height 0 of the beam spot on the scanned surface.Therefore, the deflected luminous flux is weakly convergent in the mainscanning direction, and divergent in the sub-scanning direction. Thescanning image-forming optical system 22, 24 is preferably an anamorphicoptical system having a function of placing the vicinity of thedeflection reflective surface 20A and the position of the scannedsurface 26 in a geometric optical conjugate relationship relative to thesub-scanning direction, and also has a function of converging thedeflected luminous flux as the converged luminous flux weak in the mainscanning direction onto the scanned surface.

For the rotating polygon mirror 20 functioning as the light deflectingdevice, the number of deflection reflective surfaces is preferably six,an inscribed circle radius is about 18 mm, an incident angle shown inFIG. 1 is θ=60 degrees, and a distance between rotating axis 20X andscanning image-forming optical system optical axis AX is h=7.80 mm.

Additionally, here, to make supplementary comments on the optical axisAX shown in FIG. 1, in Example 1, the lens surfaces of two lenses 22, 24constituting the scanning image-forming optical system are provided withtilt angles. The optical axis AX is considered to be a reference whenthe tilt angle is set to zero. When tilt is imparted to the reference,the actual surface direction of the lens 22, 24 is determined. Thecoaxial non-spherical surface, non-circular arc shape in the mainscanning cross section, sub non-circular arc surface and the like haveshapes specified where the tilt angle is set to zero.

A field angle of the scanning image-forming optical system is in therange of about −40.14 to about +40 degrees. Both surfaces of the lens 22are coaxial non-spherical surfaces. The lens 24 has a sub non-circularare surface on the side of the scanned surface, and a non-circular arcshape in the main scanning, cross section and a circular arc shape inthe sub-scanning, cross section on the side of the rotating polygonmirror. Data on and after the deflection reflective surface (curvatureradius is a paraxial curvature radius in the non-circular arc shape) areas follows:

Surface No. Rm Rs(0) x α n Deflection 0 ∞ ∞ 26.38 reflective surfaceLens 22 1 −100.91 −100.91 18.00 0.10 1.52441 2 −76.40 −76.40 13.06 −0.17Lens 24 3 4657.6 100.03 15.00 −0.46 1.52441 4 −159.24 −30.04 −30.04−0.46

Character “x” denotes a surface interval on the optical axis (when thetilt angle is zero), “α” denotes a tilt angle (in degrees, positive in acounterclockwise direction), and “n” denotes a refractive index of alens material (for a desired wavelength of about 780 nm in all of theExamples 1 to 3)

Surface number 1, 2 represents the coaxial non-spherical surface, whichis specified by applying each constant of the equation (A). For thesurface number 3, the non-circular arc shape in the main scanning crosssection is specified by giving each constant of the equation (B).Moreover, the shape in the sub-scanning cross section is a circular arc,but its curvature radius changes in the main scanning direction, and theshape is specified when each constant of the equation (C) or (D) isgiven. Moreover, since the surface number 4 denotes the sub non-circulararc surface, the shape in the main scanning cross section is determinedby equation (B), the curvature change of the main scanning direction inthe sub-scanning cross section is determined by equation (C), and thenon-circular arc shape in the sub scanning cross section and its changeof the main scanning direction are specified by equations (E) and (F).

Coefficients of main and sub-scanning directions of surfaces are shownin Table 1.

TABLE 1 (Example 1) Surface Main scanning direction number directioncoefficient coefficient 1 K −56.172 — Equation (A) A₄ −2.7017 × 10⁻⁶  —A₆   4.4068 × 10⁻⁵  — A₈ −4.7740 × 10⁻¹² — A₁₀   1.7929 × 10⁻⁵  — 2 K−10.639 — Equation (A) A₄ −1.6089 × 10⁻⁷  — A₆   1.0329 × 10⁻¹¹ — A₈−1.2355 × 10⁻¹⁴ — A₁₀   5.7939 × 10⁻¹⁷ — 3 K −28683 B₁ 0 Equation (B) A₁0 B₂   1.1262 × 10⁻²  Equation (C) A₃ 0 B₃ 0 A₄ −5.7527 × 10⁻⁸  B₄−3.1361 × 10⁻⁶  A₅ 0 B₅ 0 A₆ −3.0952 × 10⁻¹² B₆ −3.0627 × 10⁻¹⁰ A₇ 0 B₇0 A₈   4.0588 × 10⁻¹⁶ B₈   1.6254 × 10⁻¹⁰ A₉ 0 B₉ 0 A₁₀ −3.80114 ×10⁻¹⁹  B₁₀ −2.2486 × 10⁻¹⁶ 4* K −0.9189 B₁ −1.9122 × 10⁻⁸  Equation (B)A₁ B₂   1.0415 × 10⁻⁶  Equation (C) A₃ B₃ −8.8512 × 10⁻⁹  A₄ −7.3824 ×10⁻⁷  B₄   5.3763 × 10⁻¹¹ A₅ B₅   7.2498 × 10⁻¹² A₆   1.6824 × 10⁻¹⁰ B₆−1.1746 × 10⁻¹³ A₇ B₇ −3.5899 × 10⁻¹⁵ A₈ −3.5784 × 10⁻¹⁴ B₈   6.7888 ×10⁻¹⁷ A₉ B₉   6.9604 × 10⁻¹⁹ A₁₀   3.0590 × 10⁻¹⁸ B₁₀ −3.2046 × 10⁻²⁰

Coefficients of sub-scanning direction of surface No. 4 (subnon-circular arc surface of the lens 24 on the side of the scannedsurface) are shown in Table 2.

TABLE 2 (Example 1) 4* C₀ 1.0407 K₀ −9.4277 × 10⁻⁷  Equation (E) C₁  2.1108 × 10⁻³  K₁ −3.9882 × 10⁻⁹  Equation (F) C₂ −8.4082 × 10⁻⁴  K₂  8.4166 × 10⁻¹⁰ C₃ −2.0507 × 10⁻⁷  K₃   4.1230 × 10⁻¹³ C₄   5.8385 ×10⁻⁵  K₄ −5.8602 × 10⁻¹⁴ I₀ −4.0142 × 10⁻⁵  M₀   2.1842 × 10⁻⁶  I₁−1.7132 × 10⁻⁸  M₁   5.9053 × 10⁻⁹  I₂   3.5246 × 10⁻⁹  M₂ −1.0858 ×10⁻⁹  I₃ −1.7825 × 10⁻¹² M₃ −6.0908 × 10⁻¹³ I₄ −2.4875 × 10⁻¹³ M₄  7.8246 × 10⁻¹⁴

FIG. 3 shows in more detail the specifics of the non-arc shape asdescribed above.

FIG. 4 shows an image-surface curvature (in the left portion, a solidline shows the sub-scanning direction and a broken line shows the mainscanning direction) and constant velocity characteristic (a solid lineshows linearity and a broken line shows fθ characteristic) regarding thefirst preferred embodiment. The image-surface curvature and the constantvelocity characteristic arc both corrected remarkably satisfactorily.

FIGS. 5(a) and 5(b) show a depth curve of a beam spot diameter(variation of beam spot diameter relative to the defocusing amount ofthe beam spot) in the image height of 0, ±50 mm, ±100 mm of the beamspot in the first preferred embodiment. While FIG. 5(a) relates to themain scanning direction, FIG. 5(b) relates to the sub-scanningdirection. It is intended in the first preferred embodiment that thebeam spot diameter defined with an intensity equal to about 1/e² of linespread function is about 50 μm, and about 50 μm or less on the imagesurface. As shown in the drawing, the depth is satisfactory in eitherthe main or sub-scanning direction, and an allowable degree for positionprecision of the scanned surface is high.

EXAMPLE 2

Next, Example 2 is a specific example of the preferred embodiment shownin FIG. 2.

The luminous flux from the light source 10 or semiconductor laser iscoupled by the coupling lens 11 to form the parallel luminous flux.Therefore, the position of the natural light-converging point is at aposition of so toward the scanned surface from the deflection reflectivesurface. Accordingly, the deflected luminous flux is a parallel luminousflux in the main scanning direction, and divergent in the sub-scanningdirection. The scanning image-forming optical system 21, 23 is ananamorphic optical system having a function of placing the vicinity ofthe deflection reflective surface 19A and the position of the scannedsurface 26 in a geometric optical conjugate relationship relative to thesub-scanning direction, and also has a function of converging theparallel luminous flux in the main scanning direction onto the scannedsurface. The rotating polygon mirror 19 as the light deflecting devicepreferably has 5 deflection reflective surfaces and an inscribed circleradius of about 13 mm, and an incident angle shown in FIG. 1 is θ=60degrees, and a distance between a rotating axis 19X and a scanningimage-forming optical system optical axis AX is preferably about h=7.53mm.

In Example 2, of two lenses 21, 23 constituting the scanningimage-forming optical system, the lens surface of the lens 23 on theside of the rotating polygon mirror is provided with a shift amount(parallel movement in the main scanning direction and positive upward).The optical axis AX is considered to be a reference when the shiftamount is set to zero. When the shift amount is provided to thereference, the actual position of the lens surface is determined. Inthis case, the coaxial non-spherical surface, non-circular arc shape inthe main scanning cross section, sub non-circular arc surface and thelike have the shapes specified and have a shift amount set to zero.

The field angle of the scanning image-forming optical system is in therange of about −39.97 to about +39.97 degrees. Both surfaces of the lens21 are coaxial non-spherical surfaces. The lens 23 has a subnon-circular arc surface on the side of the rotating single-surfacemirror, and circular arc shapes (curvature radius in the sub-scanningcross section changes in the main scanning direction) in both the mainand sub-scanning cross sections on the side of the scanned surface. Thedata on and after the deflection reflective surface (curvature radius isa paraxial curvature radius in the non-circular arc shape) are asfollows:

Surface No. Rm Rs(0) x y α n Deflection reflective ∞ ∞ 25.44 surface 0Soundproof glass 1 ∞ ∞ 2.01 8 1.51433 2 ∞ ∞ 25.42 8 Lens 21 3 −312.60−312.60 31.40 1.52716 4 −82.95 −82.95 78.00 Lens 23 5 −500.0 −42.67 3.50−0.41 1.52716 6 −1000.0 −23.32 141.50

Character “x” denotes a surface interval on the optical axis (when theshift is zero), “y” denotes the shift amount, and “α” denotes the tiltangle of the soundproof glass (in degrees).

Since both of the surface numbers 3, 4 are preferably coaxialnon-spherical surfaces, they are specified when given the constants ofthe equation (A). For the surface number 6, the curvature radius of thecircular arc shape in the sub-scanning cross section changes in the mainscanning direction, and the surface is specified when given theconstants of the equation (C) or (D). Moreover, since the surface number5 denotes the sub non-circular arc surface, the shape in the mainscanning cross section is determined by equation (B), the curvaturechange of the main scanning direction in the sub-scanning cross sectionis determined with by (C), and the non-circular arc shape in thesub-scanning cross section and its change of the main scanning directionarc specified by equations (E) and (F).

Coefficients of main and sub-scanning directions of surfaces are shownin Table 3.

TABLE 3 (Example 2) Main scanning direction Surface number coefficientdirection coefficient 3 K 2.6671 — Equation (A) A₄   1.7857 × 10⁻⁷  — A₆−1.0807 × 10⁻¹² — A₈ −3.1812 × 10⁻¹⁴ — A₁₀   3.7403 × 10⁻¹⁸ — 4 K 0.0193Equation (A) A₄   2.5031 × 10⁻⁷  — A₆   9.6058 × 10⁻¹² — A₈   4.5447 ×10⁻¹⁵ — A₁₀ −3.0343 × 10⁻¹⁸ — 5* K −71.7319 B₁ 0 Equation (B) A₁ 0 B₂−4.72602Z10⁻⁷  Equation (C) A₃ 0 B₃ −8.38263Z10⁻¹⁰ A₄   4.3256 × 10⁻⁸ B₄   9.04894Z10⁻¹¹ A₅ 0 B₅   4.37405Z10⁻¹³ A₆ −5.9729 × 10⁻¹³ B₆−6.01229Z10⁻¹⁵ A₇ 0 B₇ −6.46929Z10⁻¹⁷ A₈ −1.2819 × 10⁻¹⁶ B₈  2.55750Z10⁻²⁰ A₉ 0 B₉   3.04068Z10⁻²¹ A₁₀   5.7297 × 10⁻²¹ B₁₀  2.36799Z10⁻²³ 6 K 0 B₁ 0 Equation (B) A₁ 0 B₂ −1.5557 × 10⁻⁴  Equation(C) A₃ 0 B₃ 0 A₄ 0 B₄ −7.64287 × 10⁻¹⁰  A₅ 0 B₅ 0 A₆ 0 B₆   1.29011 ×10⁻¹² A₇ 0 B₇ 0 A₈ 0 B₈ −8.88372 × 10⁻¹⁷ A₉ 0 B₉ 0 A₁₀ 0 B₁₀ −6.79872 ×10⁻²¹

Coefficients of sub-scanning direction of surface No. 5 (subnon-circular are surface of lens 23 on the side of the rotating polygonmirror) are shown in Table 4.

TABLE 4 (Example 2) 5* C₀ 2.5821 K₀ −3.77145 × 10⁻⁷  Equation (E) C₁−2.99305 × 10⁻⁴  K₁   4.72035 × 10⁻¹² Equation (F) C₂ −6.20612 × 10⁻⁵ K₂   2.65693 × 10⁻¹¹ C₃ 0 K₃ −1.69246 × 10⁻¹⁵ C₄ 0 K₄ −1.20928 × 10⁻¹⁵I₀ −3.02237 × 10⁻⁵  M₀ −1.71892 × 10⁻⁹  I₁ −2.75077 × 10⁻¹⁰ M₁ −8.23793× 10⁻¹³ I₂   6.59149 × 10⁻¹¹ M₂ −5.52334 × 10⁻¹⁵ I₃   2.06446 × 10⁻¹⁴ M₃  1.23174 × 10⁻¹⁶ I₄   7.98792 × 10⁻¹⁸ M₄   2.17562 × 10⁻¹⁷

FIG. 6 shows an image-surface curvature and constant velocitycharacteristic regarding Example 2 in the same manner as FIG. 3. Theimage-surface curvature and the constant velocity characteristic areboth corrected remarkably satisfactorily.

FIGS. 7(a) and 7(b) show a depth curve of a beam spot diameter at animage height of 0, ±100 mm, ±150 mm of the beam spot in Example 2. WhileFIG. 7(a) relates to the main scanning direction, FIG. 7(b) relates tothe sub-scanning direction. The beam spot diameter intended in theembodiment 2 is about 30 μm with the beam spot diameter having anintensity of about 1/e² of line spread function. As shown in thedrawing, the depth is satisfactory in either main or sub-scanningdirection, and an allowable degree for position precision of the scannedsurface is high.

FIGS. 8(a)-8(e) show wave aberrations in Example 2. While FIG. 8(a)shows a wave aberration in the image height of ±150 mm, FIG. 8(b) showsa wave aberration in the image height of ±100 mm, FIG. 8(c) denotes animage height of 0 mm, and FIGS. 8(d) and 8(c) show wave aberrations inthe image heights of −100 mm, −150 mm, respectively. Additionally, inthis case, the aperture 13 for shaping beams which has a substantiallyrectangular opening is used.

As seen from FIGS. 8(a)-8(e), in Example 2, the wave aberration issubstantially completely corrected at the image height 0 of the beamspot, but slightly insufficiently corrected for four corners with theimage height of +150 mm. The deterioration of the wave aberration offour corners on a pupil as shown in FIG. 8(a) influences the shape/spotdiameter of the beam spot. In Example 2, as described above, asubstantially rectangular shape (substantially rectangular shape havinglong sides in the main scanning direction, and short sides in thesub-scanning direction) is used as the opening shape of the aperture 13for shaping beams is used. To remove the influence of the deteriorationof the wave aberration on four corners on the pupil onto the beam spotas shown in FIG. 8(a), an opening shape of the aperture 13 preferably issubstantially rectangular or elliptical and includes cut off four-cornerportions of the main/subs-canning direction of the coupled luminous fluxso as to define a substantially rectangular or elliptical shapeincluding four rounded four corners.

In Example 2, when the substantially rectangular or elliptical shapewith four rounded corners is used as the opening shape of the aperture13, wave aberrations on the pupil in the beam spot image heights of 0,±100 mm, ±150 mm are shown. Specifically, FIG. 9(a) shows a waveaberration at the image height of 150 mm, FIG. 9(b) shows a waveaberration at the image height of 100 mm, FIG. 9(c) shows a waveaberration at the image height of 0, FIG. 9(d) shows a wave aberrationat the image height of −100 mm, and FIG. 9(e) shows a wave aberration inthe image height of ±150 mm.

As compared with the wave aberrations shown in FIGS 8(a)-8(e), RMS waveaberrations arc remarkably improved on the pupil surface. In this case,FIGS. 12(a) and 12(b) show beam spot diameter depth curves at the beamspot image heights of 0, ±100 mm, ±150 mm. While FIG. 12(a) relates tothe main scanning direction, FIG. 12(b) relates to the sub-scanningdirection. As compared with the case of FIGS. 6(a) and 6(b) (theaperture having a substantially rectangular opening is used), especiallythe depth allowance of the sub-scanning direction can be enlarged.

FIG. 10 shows changes of the paraxial curvature of the sub-scanningcross section of the sub non-circular arc surface (the surface on theside of rotating polygon mirror of the lens on the side of the scannedsurface with the surface number 5) in the main scanning direction inExample 2. As shown in FIG. 10, the paraxial curvature in thesub-scanning cross section changes in the main scanning direction, andthe curvature change is asymmetric in the main scanning direction andhas two or more extreme values.

EXAMPLE 3

Example 3 is also a specific example of the preferred embodiment shownin FIG. 2.

The luminous flux from the light source 10 or semiconductor laser iscoupled by the coupling lens 11 to form the parallel luminous flux.Therefore, the deflected luminous flux is a parallel luminous flux inthe main scanning direction, and divergent in the sub-scanningdirection. The scanning image-forming optical system 21, 23 is ananamorphic optical system having a function of placing the vicinity ofthe deflection reflective surface 19A and the position of the scannedsurface 26 in a geometric optical conjugate relationship relative to thesub-scanning direction, and also has a function of converging thedeflected luminous flux as the parallel luminous flux in the mainscanning direction onto the scanned surface 26.

The rotating polygon mirror 19 functioning as the light deflectingdevice preferably has five deflection reflective surfaces and aninscribed circle radius of about 13 mm, and the incident angle shown inFIG. 1 is θ=60 degrees, and the distance between the rotating axis andthe scanning image-forming optical system optical axis AX is abouth=5.22 mm.

In Example 3, two lenses 21, 23 constituting the scanning image-formingoptical system are provided with no tilt angle/shift amount.

The field angle of the scanning image-forming optical system is in therange of about −42 to about +42 degrees. The lens 21 has a coaxialnon-spherical surface on the side of the rotating polygon mirror, and anon-circular arc shape in the main scanning cross section and a circulararc shape in the sub-scanning cross section on the side of the scannedsurface, while the curvature radius changes in the main scanningdirection. The lens 23 has a sub non-circular are surface on the side ofthe rotating polygon mirror, and a toroidal surface on the side of thescanned surface. The data on and after the deflection reflective surface(curvature radius is a paraxial curvature radius in the non-circular arcshape) are as follows:

Surface No. Rm Rs(0) x y α n Deflection reflective ∞ ∞ 25.44 1.588surface 0 Soundproof glass 1 ∞ ∞ 2.01 8 1.51433 2 ∞ ∞ 25.42 8 Lens 21 3−312.60 −312.60 31.40 1.52716 4 −82.95 104.02 78.00 Lens 23 5 −500.0−63.50 3.50 1.52716 6 −1000.0 −23.38 143.38

Shift “y” means that the optical system on and after the deflectionreflective surface entirely shifts only by 1.588, and “α” denotes thetilt angle (in degrees) of the soundproof glass.

The surface number 3 is a coaxial non-spherical surface specified whengiven the constants of the equation (A). For the surface number 4, thenon-circular arc shape in the main scanning cross section is specifiedby equation (B), and the change of the main scanning direction in thecurvature radius in the sub-scanning cross section is specified byequation (C). Since the surface number 5 denotes the sub non-circulararc surface, the shape in the main scanning cross section is determinedby equation (B), the curvature change of the main scanning direction inthe sub-scanning cross section is determined by equation (C), and thenon-circular arc shape in the sub-scanning cross section and its changeof the main scanning direction are specified by equations (E) and (F).

Coefficients of main and sub-scanning directions of surfaces are shownin Table 5.

TABLE 5 (Example 3) Surface Main scanning direction number directioncoefficient coefficient 3 K 2.6671 — Equation (A) A₄   1.7857 × 10⁻⁷  —A₆ −1.0807 × 10⁻¹² — A₈ −3.1812 × 10⁻¹⁴ — A₁₀   3.7403 × 10⁻¹⁸ — 4 K0.0193 B₂ −2.1855 × 10⁻⁷  Equation (B) A₄   2.5031 × 10⁻⁷  B₄ −6.8348 ×10⁻¹⁰ Equation (C) A₆   9.6058 × 10⁻²  B₆   1.9548 × 10⁻¹³ A₈   4.5447 ×10⁻¹⁵ B₈   8.6565 × 10⁻¹³ A₁₀ −3.0343 × 10⁻¹⁸ B₁₀ −1.3677 × 10⁻²⁰ B₁₂−3.2528 × 10⁻²⁴ B₁₄   4.7205 × 10⁻²⁸ B₁₆   3.9957 × 10⁻³¹ B₁₈ −5.8552 ×10⁻³⁵ 5* K −71.7319 B₁ −9.5132-07 Equation (B) A₁ 0 B₂ −1.0179-06Equation (C) A₃ 0 B₃   2.9721-10 A₁   4.3256 × 10⁻⁸  B₄   7.5379-11 A₅ 0B₅ −2.9473-14 A₆ −5.9729 × 10⁻¹³ B₆  12 5.4844-16 A₇ 0 B₇   4.1734-19 A₈−1.2819 × 10⁻¹⁶ B₈ −1.9406-19 A₉ 0 B₉   1.0447-22 A₁₀   5.7297 × 10⁻²¹B₁₀ −1.2851-23 B₁₁   2.1339-27 B₁₂   5.0995-29 B₁₃ −4.6561-31 B₁₄  7.5063-32 B₁₅ −4.4273-35 B₁₆   5.8948-36 B₁₇   2.6911-39 B₁₈−3.4755-40

Coefficients of sub-scanning direction of surface No. 5 (subnon-circular arc surface of lens 23 on the side of the rotating polygonmirror) are shown in Tables 6 and 7.

TABLE 6 (Example 3) 5* C₀ −0.10052+02   K₀ −0.54249−05   Equation (E) C₁0.10456−01 K₁ 0.27589−08 Equation (F) C₂ 0.16043−01 K₂ 0.40441−08 C₃−0.38810−04   K₃ −0.10214−10   C₄ −0.10795−04   K₄ −0.30236−11   C₅0.24649−07 K₅ 0.74022−14 C₆ 0.34813−08 K₆ 0.10175−14 C₇ −0.79177−11   K₇−0.22278−17   C₈ −0.62576−12   K₈ −0.18297−18   C₉ 0.14305−14 K₉0.35070−21 C₁₀ 0.68155−16 K₁₀ 0.19279−22 C₁₁ −0.15016−18   K₁₁−0.31042−25   C₁₂ −0.46481−20   K₁₂ −0.12245−26   C₁₃ 0.90675−23 K₁₃0.15609−29 C₁₄ 0.19560−24 K₁₄ 0.45816−31 C₁₅ −0.29175−27   K₁₅−0.41677−34   C₁₆ −0.46741−29   K₁₆ −0.91912−36   C₁₇ 0.38721−32 K₁₇0.45899−39 C₁₈ 0.48737−34 K₁₈ 0.74711−41 I₀ 0.50108−05 M₀ 0.60483−06 I₁−0.71522−08   M₁ −0.60493−10   I₂ −0.94329−08   M₂ −0.38193−09   I₃0.29968−10 M₃ 0.44521−12 I₄ 0.87187−11 M₄ 0.25695−12 I₅ −0.28570−13   M₅−0.20044−15   I₆ −0.35454−14   M₆ −0.75042−16   I₇ 0.10163−16 M₇−0.24566−20   I₈ 0.78408−18 M₈ 0.10314−19 I₉ −0.18615−20   M₉ 0.12901−22I₁₀ −0.10350−21   M₁₀ −0.59018−24   I₁₁ 0.19267−24 M₁₁ −0.25130−26   I₁₂0.84093−26 M₁₂ −0.88063−29   I₁₃ −0.11391−28   M₁₃ 0.20958−30 I₁₄−0.41267−30   M₁₄ 0.29035−32 I₁₅ 0.35905−33 M₁₅ −0.82447−35   I₁₆0.11223−34 M₁₆ −0.13775−36   I₁₇ −0.46817−38   M₁₇ 0.12546−39 I₁₈−0.12986−39   M₁₈ 0.21833−41

TABLE 7 (Example 3) 5* O₀ −0.22979-07 O₁₀ −0.59495-25 Equation (E) O₁−0.46238-11 O₁₁   0.15648-27 Equation (F) O₂   0.11760-10 O₁₂  0.82774-29 O₃ −0.41106-14 O₁₃ −0.11727-31 O₄ −0.62792-14 O₁₄−0.56109-33 O₅ −0.14430-17 O₁₅   0.43914-36 O₆   0.10515-17 O₁₆  0.19030-37 O₇   0.32077-20 O₁₇ −0.65284-41 O₈   0.11144-21 O₁₈−0.25859-42 O₉ −0.10653-23

FIG. 11 shows an image-surface curvature and constant velocitycharacteristic regarding the Example 3 in the same manner as FIG. 4. Theimage-surface curvature and the constant velocity characteristic areboth corrected remarkably satisfactorily.

Non-circular arc amounts of the non-circular arc shape in thesub-scanning cross section of the surface 5 as the sub non-circular arcsurface in Example 3 (lens surface on the side of a light deflector ofthe lens on the side of the scanned surface), i.e., deviation amountsfrom a circular arc (in the unit of mm) are shown in Table 8.

TABLE 8 (Example 3) Y coordinate mm/ Z coordinate mm 0.0 0.5 1.0 1.5 2.02.5 3.0 3.5 4.0 120 0.000 0.000 −0.001 −0.004 0.011 0.106 0.400 0.9451.190 110 0.000 0.000 −0.001 0.000 0.027 0.140 0.429 0.924 1.311 1000.000 0.000 0.000 0.010 0.061 0.211 0.521 1.039 1.895 90 0.000 0.0000.001 0.015 0.081 0.259 0.589 1.091 2.019 80 0.000 0.000 0.000 0.0150.087 0.277 0.619 1.153 2.359 70 0.000 0.000 −0.001 0.012 0.084 0.2740.614 1.209 2.984 60 0.000 0.000 −0.001 0.011 0.081 0.269 0.600 1.2253.395 50 0.000 0.000 −0.001 0.013 0.086 0.277 0.608 1.250 3.550 40 0.0000.000 0.000 0.016 0.094 0.289 0.628 1.306 3.735 30 0.000 0.000 0.0000.017 0.095 0.288 0.626 1.339 4.000 20 0.000 0.000 −0.001 0.011 0.0830.268 0.594 1.304 4.170 10 0.000 0.000 −0.004 0.002 0.066 0.244 0.5561.238 4.182 0 0.000 −0.001 −0.005 −0.003 0.057 0.234 0.543 1.212 4.158−10 0.000 0.000 −0.004 0.002 0.066 0.246 0.563 1.247 4.165 −20 0.0000.000 −0.001 0.011 0.084 0.272 0.600 1.304 4.105 −30 0.000 0.000 0.0010.018 0.098 0.293 0.631 1.330 3.908 −40 0.000 0.000 0.001 0.018 0.0980.294 0.638 1.316 3.692 −50 0.000 0.000 −0.001 0.013 0.087 0.280 0.6201.280 3.594 −60 0.000 0.000 −0.002 0.009 0.080 0.268 0.600 1.235 3.477−70 0.000 0.000 −0.001 0.011 0.082 0.271 0.600 1.186 3.101 −80 0.0000.000 0.000 0.014 0.086 0.275 0.613 1.166 2.639 −90 0.000 0.000 0.0000.012 0.077 0.258 0.595 1.141 2.389 −100 0.000 0.000 −0.002 0.005 0.0580.208 0.470 0.808 1.459 −110 0.000 0.000 −0.003 −0.002 0.032 0.126 0.192−0.125 −1.464 −120 0.000 0.000 −0.004 −0.007 0.014 0.057 −0.071 −1.047−4.370

The non-circular arc amount changes asymmetrically to the lens opticalaxis in accordance with the position in the main scanning direction ofthe sub-scanning cross section. By setting the non-circular arc amountin this manner, the wave aberrations on the pupil are corrected for allimage heights, and the influence of optical sag is eliminated, so that asatisfactory small-diameter beam spot can be formed.

Regarding the above-mentioned Examples 1 to 3, parameter values ofcondition (1), and condition (4) will be described below.

Condition (1)

Example 1: |β₀|=2.51

Example 2: |β₀|=0.78

Example 3: |β₀|=0.73

Since the condition (1) is satisfied in Examples 2, 3, but not satisfiedin Example (1), Example 1 has a larger limitation to the reduction ofthe beam spot diameter as compared with examples 2, 3.

condition (4 )

Example 1: Fs/W=0.131/216=0.0006

Example 2: Fs/W=0.131/300=0.0001

Example 3: Fs/W=0.137/320=0.0004

Examples 1 to 3 preferably all satisfy condition (4), so that a verystable small-diameter beam spot is obtained by suppressing the variationof the sub-scanned image-surface curvature. Additionally, Examples 1 to3 all preferably satisfy condition (2). More specifically, in eachexample of preferred embodiments, by using two or more surfaces in whichthe paraxial curvature in the sub-scanning cross section changes inaccordance with the main scanning direction, the front/rear main pointposition is set in a desired position, the magnification of each imageheight is kept constant, and a stable beam spot is obtained. In Examples1, 2, the main point position is arbitrarily set by bending incident andemission side surfaces of the lens on the side of the scanned surface,to achieve a constant lateral magnification. Moreover, in Example 3, themain point position is arbitrarily set by bending the emission-sidesurface of the lens on the side of the rotating polygon mirror, and theincident-side surface on the side of the scanned surface, to achieve aconstant lateral magnification.

In Example 2, as shown in FIG. 9, the paraxial curvature in thesub-scanning cross section of the lens surface on the side of therotating polygon mirror (incident side) of the lens on the side of thescanned surface of the scanning image-forming optical system changesasymmetrically in the main scanning direction, and three extreme valuepositions a, b, c are provided. For these a, b, c positions, calculationof parameters of condition (3) provide following:

point a: |(he)/(hmax)|=|(−65)/(−90)|=0.72

point b: |(he)/(hmax)|=|(0)/(44.8)|=0

point c: |(he)/(hmax)|=|(+62)/(+90)|=0.69

Of the three extreme values, the extreme values by which theimage-surface curvature is effectively corrected are points a and cwhich satisfy the condition (3).

Since the scanning image-forming optical system of preferred embodimentsof the present invention includes special surface shapes as describedabove, manufacturing using plastic as a material for the lenses of thescanning image-forming optical system is suitable.

As described above, according to the scanning image-forming opticalsystem of preferred embodiments of the present invention, because thewave aberration is effectively corrected using the sub non-circular arcsurface, a small-diameter beam spot of about 50 μm or less can bereliably achieved. Moreover, in the optical scanning device of preferredembodiments of the present invention, the scanning image-forming opticalsystem can be used to realize a greatly increased writing density and avery small-diameter beam spot having a uniform diameter.

While preferred embodiments of the invention have been disclosed,various modes of carrying out the principles disclosed herein arecontemplated as being within the scope of the following claims.Therefore, it is understood that the scope of the invention is not to belimited except as otherwise set forth in the claims.

What is claimed is:
 1. An optical scanning apparatus comprising: a lightsource for outputting light; a first lens system arranged to receive thelight output from the light source and to transmit a light fluxtherefrom; an optical deflector arranged to receive the light flux fromthe first lens system and having a deflecting reflective plane todeflect the light flux from a surface therefrom; and a second lenssystem arranged to receive the light flux deflected from the opticaldeflector and to condense the deflected luminous flux into an opticalbeam spot on a surface to be scanned so as to form images having imageheights, the luminous flux condensed by the second lens system into theoptical beam spot including an optical beam waist, the second lenssystem including a scanning and image forming element including at leastone surface including a plurality of portions each having a non-arcshape in a sub-scanning direction such that at least two of the non-arcshapes are different from each other and such that an effective writingwidth W and a width Fs of the sub-scanned image-surface curvaturelocated within the effective writing width satisfies the conditionFs/W<0.005.
 2. An optical scanning apparatus according to claim 1,wherein the second lens system includes at least two lens elementsproviding four lens surfaces, each of the four lens surfaces having thenon arc shape in the main scanning direction.
 3. An optical scanningapparatus according to claim 1, wherein the second lens system includesat least two lens elements providing four lens surfaces, at least threeof the four lens surfaces having the non arc shape in the main scanningdirection.
 4. An optical scanning apparatus according to claim 1,wherein the second lens system includes at least two lens elementsproviding four lens surfaces, at least two of the four lens surfaceshaving the non arc shape in the sub-scanning direction and each of thenon arc shapes of the at least two surfaces is shaped such that a radiusof curvature in a sub-scanning cross-section changes in a directioncorresponding to the main scanning direction such that a curvaturecenter line plotting a curvature center in the sub-scanningcross-section of the at least one surface in the direction correspondingto the main scanning direction is a curve which is different from thenon arc shape in the deflecting reflective plane.
 5. An optical scanningapparatus according to claim 1, wherein the second lens system includesat least three lens elements.
 6. An optical scanning apparatus accordingto claim 1, wherein the second lens system includes only one lenselement.
 7. An optical scanning apparatus according to claim 1, whereinthe second lens system includes at least two lens elements.
 8. Anoptical scanning apparatus according to claim 1, wherein the imagesformed on the surface to be scanned have a writing density of about 600dots per inch to about 1200 dots per inch.
 9. An optical scanningapparatus according to claim 1, wherein the images formed on the surfaceto be scanned have a writing density of about 1200 dots per inch toabout 2400 dots per inch.
 10. An optical scanning apparatus according toclaim 1, wherein the images formed on the surface to be scanned have awriting density of greater than about 2400 dots per inch.
 11. An opticalscanning apparatus according to claim 1, wherein the light source isconstructed to emit multiple light beams.
 12. An optical scanningapparatus according to claim 1, wherein the second lens system includesat least one lens having a sub noncircular arc surface having a shapethat is configured to align the beam waist position with a geometricoptic image forming position.
 13. An optical scanning apparatusaccording to claim 1, wherein the scanning and image forming elementcomprises an anamorphic optical system which is arranged such that thedeflecting reflective surface and the scanned surface position have ageometric optical conjugate relationship for the sub-scanning direction.14. An optical scanning apparatus according to claim 1, wherein alateral magnification β₀ on the optical axis and a lateral magnificationβ_(h) at an arbitrary image height h, both in the sub-scanningdirection, satisfies a condition 0.93 <|β_(h)/β₀<1.07.
 15. An opticalscanning apparatus according to claim 1, wherein the non-arc shape ofthe plurality of portions of the at least one surface are arranged suchthat the beam waist of the entire light flux is located at the surfaceto be scanned for all image heights.
 16. An image forming apparatuscomprising: a light source for outputting light; a first lens systemarranged to receive the light output from the light source and totransmit a light flux therefrom; an optical deflector arranged toreceive the light flux from the first lens system and having adeflecting reflective plane to deflect the light flux from a surfacetherefrom; and a second lens system arranged to receive the light fluxdeflected from the optical deflector and to condense the deflectedluminous flux into an optical beam spot on a surface to be scanned so asto form images having image heights, the luminous flux condensed by thesecond lens system into the optical beam spot including an optical beamwaist, the second lens system including a scanning and image formingelement including at least one surface including a plurality of portionseach having a non-arc shape in a sub-scanning direction such that atleast two of the non-arc shapes are different from each other and suchthat an effective writing width W and a width Fs of the sub-scannedimage-surface curvature located within the effective writing widthsatisfies the condition Fs/W<0.005.
 17. A method of forming a lenssystem for an optical scanning apparatus for optically scanning asurface to be scanned by deflecting a luminous flux emitted from a lightsource at equiangular velocity via an optical deflector so as totransmit the deflected luminous flux through the lens system and tocondense the deflected luminous flux into an optical beam spot on thesurface to be scanned so as to form images having image heights, theluminous flux condensed by the lens system into the optical beam spotincluding an optical beam waist, the method comprising: forming ascanning and image forming element to have at least one surface having aplurality of portions each of which contains a non-arc shape in asub-scanning direction such that all beam spot diameters are within arange for all image heights and such that an effective writing width Wand a width Fs of the sub-scanned image-surface curvature located withinthe effective writing width satisfies the condition Fs/W<0.005.